Rigorous derivation of nonlinear plate theory and geometric rigidity
[Dérivation rigoureuse de la théorie non linéaire des plaques et rigidité géométrique]
Friesecke, Gero1 ; Müller, Stefan2 ; James, Richard D.3
1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
2 Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103 Leipzig, Germany
3 Department of Aerospace Engineering and Mechanics, 107 Akerman Hall, University of Minnesota, Minneapolis, MN 55455, USA
Comptes Rendus. Mathématique, Tome 334 (2002) no. 2, pp. 173-178.

Nous montrons que la théorie non linéaire des plaques émerge comme Γ-limite de la théorie de l'élasticité tridimensionnelle. La démonstration repose sur un résultat de rigidité pour des fonctions v:(0,1) 3 →ℝ 3 . Nous montrons que la distance L2 de ∇v d'une rotation est bornée par un multiple de la distance L2 à l'ensemble SO(3) des rotations.

We show that nonlinear plate theory arises as a Γ-limit of three-dimensional nonlinear elasticity. A key ingredient in the proof is a sharp rigidity estimate for maps v:(0,1) 3 →ℝ 3 . We show that the L2 distance of ∇v from a single rotation is bounded by a multiple of the L2 distance from the set SO(3) of all rotations.

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Accepté le :
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DOI : 10.1016/S1631-073X(02)02133-7
Friesecke, Gero 1 ; Müller, Stefan 2 ; James, Richard D. 3

1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
2 Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103 Leipzig, Germany
3 Department of Aerospace Engineering and Mechanics, 107 Akerman Hall, University of Minnesota, Minneapolis, MN 55455, USA 
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Friesecke, Gero; Müller, Stefan; James, Richard D. Rigorous derivation of nonlinear plate theory and geometric rigidity. Comptes Rendus. Mathématique, Tome 334 (2002) no. 2, pp. 173-178. doi : 10.1016/S1631-073X(02)02133-7. http://www.numdam.org/articles/10.1016/S1631-073X(02)02133-7/

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